\documentclass[]{article}
\setlength{\pdfpagewidth}{8.5in} \setlength{\pdfpageheight}{11in}

\usepackage{natbib}
           \setlength{\bibhang}{.3in}
           \setlength{\bibsep}{.05in plus0.05in minus0.05in}
           \makeatletter
           \renewcommand{\bibsection}{
           \begin{center}
           \section*{\refname\@mkboth{\MakeUppercase{\refname}}
               {\MakeUppercase{\refname}}}
           \end{center}
           }


	\bibliographystyle{jf}
	
	\begin{document}


 \begin{table}[htbp] 
  \caption{\bf Pricing Ability:  Local-PCA vs Constant-PCA  ($T=120$)}
\medskip
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccccccccc}
  \multicolumn{12}{c}{\bf Pricing Ability:  SDF }   \\  \hline 
  \multicolumn{12}{c}{Out-of-Sample}   \\  \hline 
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
 \scriptsize{  constant-PCA/local-PCA (bal)}  &
    0.960 & 1.153 & 1.076 & 0.835 & 0.742 & 0.704 & 0.687 & 0.644 & 0.625 & 0.597 & 0.570 \\
  \scriptsize{  constant-PCA/local-PCA (unbal)}  &
 5.295 &  2.634 & 3.250  & 13.605 & 2.449 & 2.646 & 2.443 & 2.010  & 1.904 & 1.993 & 2.081 \\
   \multicolumn{12}{c}{In-Sample}   \\  \hline
 \# of factors:   &  $\tilde{r}$ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
 \scriptsize{  constant-PCA/local-PCA (bal)}  &
 105.86 &   3.372 & 12.28 & 362.7 & 10.74  & 4.863  & 3.198  & 2.124 & 1.635 &  1.408  & 1.214 \\
  \scriptsize{  constant-PCA/local-PCA (unbal)}   &
 1.729 &   1.265  & 1.709   & 2.010  & 2.127   & 2.497  & 3.367 &  4.618 &  6.863 &  12.84 &  98.09 \\
 \multicolumn{12}{c}{\bf Pricing Ability:  Risk Premia }   \\  \hline
 \multicolumn{12}{c}{Out-of-Sample}   \\  \hline
 \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  \scriptsize{  constant-PCA/local-PCA (bal)} &
 0.501 & 1.126 & 1.145 & 1.162 & 1.169  &1.169 & 1.159 & 1.148 & 1.135 & 1.120 & 1.106 \\
   \scriptsize{  constant-PCA/local-PCA (unbal)}  &
  4.671 &   2.741 &  3.362  & 4.245 &   6.132 &  9.141 & 16.16 & 55.43 & 43.90 & 16.041 & 9.387 \\
    \multicolumn{12}{c}{In-Sample}   \\  \hline
 \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  \scriptsize{  constant-PCA/local-PCA (bal)}     &
 1.494 &  3.014 & 1.853 & 1.671  & 1.298 & 1.078  & 0.917  & 0.862  & 0.785 & 0.689 & 0.653 \\
   \scriptsize{  constant-PCA/local-PCA (unbal)}   &
2.399 & 1.717 &  2.794 & 3.183   & 2.399  & 3.120  & 3.519  & 4.429  & 6.633 & 10.23
& 4.124 \\ \hline \hline
    \end{tabular}
           \parbox{7.5in}{\footnotesize{Each entry of this table represents the ratio between the pricing statistics for the constant-PCA over the local-PCA for both the balanced and unbalanced panel cases:  ratios (in absolute value)  of  the time-averages (across $480-2T$ windows) of the  bias-adjusted
  SDF pricing statistics $ N^{-1}  \sum_{i=1}^N \tilde{ h}_{i}^2  $  (top two panels)       
        and of the  bias-adjusted
 risk premia pricing statistics  $ N^{-1} \sum_{i=1}^N \tilde{ \delta}_{i}^2  $
          (bottom two panels),  described in Theorem~\ref{Ttest}.
        Panels two and fourth report the in-sample results and panels one and three report the  out-of-sample  results,
    correspondingly to  the (time-varying) estimated number of factors $\tilde{r}$   (column two), described in Section~\ref{emp_number_tv}, and to the values one to $10 $ (columns 3-12), respectively.  The local-PCA is evaluated with $T=24$. Data are monthly from Jan 1980 until Dec 2019.}}
\label{Table1OAT120}
\end{table}


\bibliography{Biblio_PCA}


\end{document}
